I’m trying to find the minimum of a function with 3 variables constrained to a few restrictions with 5 variables applying the penalty method in order to optimise it using the steepest descent method with a backtracking algorithm to compute progressively the alphas.

The function and restrictions are the following:

min h(A, C, E) = 5.35*C^2+0.83*A*E+37.29*A-40.14 

subject to:

0 <= 85.33+0.0056*B*C+0.00026*A*D-0.0022*C*E <= 92 90 <= 80.51+0.0071*B*E+0.0029*A*B+0.0021*C^2 <= 110 20 <= 9.3+0.0047*C*E+0.0012*A*C+0.0019*C*D <= 25 78 <= A <= 102 33 <= B <= 45 27 <= C <= 45 27 <= D <= 45 27 <= E <= 45 

(my question is more from a theoretical point of view)

The remaining whole function to minimise will therefore be:

f(A, B, C, D, E)= h (A, C, E) + c*P(A, B, C, D, E) 

being P(A, B, C, D, E)=0.5*sum_i(max(0,gi)^2)

Now, after computing the function to minimise applying the previous formulas in the penalty formula, we need to calculate the gradient vector, which will be:

g[0] = -(0.83*E+37.29); g[1] = 0; g[2] = -2*5.35; g[3] = 0; g[4] = -0.83*A; 

The issue I have is with the steepest direction – it doesn’t tell variables B and D where to go and they will obviously remain in the initial conditions numbers regardless whether they are within the restriction parametres or not. How should I proceed with that?